[en] Nonlinear evolution of single-mode high frequency instabilities (ω approximately equal to ksub(parallel)vsub(b)) excited by a parallel-flow high-current relativistic electron beam in a magnetized plasma is investigated. Fairly general dimensionless equations are derived. They describe both the temporal and the spatial evolution of the amplitude and phase of the fundamental wave. Numerically, the special case of excitation of the linearly most unstable mode is solved in detail assuming that wave energy dissipation is negligible. The strength of interaction and the relativistic properties of the beam are fully respected by a single parameter lambda. The value of lambda ensuring the optimum efficiency of the wave excitation as well as the efficiency of the self-acceleration of some beam electrons at higher values of lambda > 1 are determined in the case of a fully compensated relativistic beam. Finally, the effect of the return current dissipation is included (phenomenologically) into the theoretical model, its role for the beam-plasma interaction being checked numerically. (author)